Damped frequency response apparatus, systems, and methods

ABSTRACT

Apparatus, systems, and methods are disclosed to solve an equation associated with a structure of the form {−ω 2 M R +iωB R +[(1+iγ)K R +i(K 4R )]}Y=F R , wherein ω is a time-harmonic excitation frequency, M R  is a reduced form of a symmetric mass matrix M, B R  is a reduced form of a viscous damping matrix B, γ is a scalar global structural damping coefficient, K R  is a reduced form of a symmetric stiffness matrix K, K 4R  is a reduced form of a symmetric structural damping matrix K 4  representing local departures from γ, F R  is a reduced form of a matrix F, and Y is a matrix having a plurality of vectors corresponding to the plurality of load vectors acting on the structure and included in the matrix F.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation application under 35 U.S.C. 111(a) ofPCT/US03/30150, filed Sep. 23, 2003, which claims priority benefit ofU.S. Provisional Application No. 60/412,895, filed Sep. 23, 2002, whichapplications are incorporated herein by reference.

TECHNICAL FIELD

Various embodiments relate generally to apparatus, systems, and methodsfor vibration analysis of various structures, including vehicles.

BACKGROUND

Using nomenclature that is well known to those of ordinary skill in theart of industrial vibration analysis, the frequency response problem fordamped structures discretized by the finite element method can bedefined by Equation (1), having the general form:{−ω² M+iωB+[(1+iγ)K+i(K ₄)]}X=F.  (1)Here ω is the radian frequency of the time-harmonic excitation andresponse, and i=√(−1). M is the mass matrix, which may be symmetric. Bis the viscous damping matrix, which may be nonsymmetric if gyroscopiceffects are modeled. It is important to note that if the structuresbeing modeled are automobiles, the matrix B may be of very low rank(e.g., less than about 50, including zero) because B's rank issubstantially equal to the number of viscous damping elements, which caninclude shock absorbers and engine mounts. If gyroscopic effects aremodeled, the rank of B is not ordinarily greatly increased.

The scalar γ is a global structural damping coefficient. K is thesymmetric stiffness matrix. K₄ is a structural damping matrix that maybe symmetric and represents local departures from the global structuraldamping level represented by γ.

X is a matrix of displacement vectors to be determined in the frequencyresponse analysis. Each vector in X represents a response of thestructure to the corresponding load vector in the matrix F. The matricesM, B, K and K₄ may be real-valued and sparse, with millions of rows andcolumns. The response matrix X can be dense and complex-valued, althoughonly a small number of rows in X, associated with specific degrees offreedom for the structure, may be of interest.

In automotive applications, the frequency response problem can be solvedat hundreds of frequencies to obtain frequency response functions over abroad frequency range. Direct or iterative solutions at each frequency,using a different coefficient matrix, is not usually feasible. Thepractical approach, therefore, has been to approximate the solutionusing the subspace of undamped natural modes of vibration having naturalfrequencies lower than a specified cutoff frequency. Because the numberof modes m is usually much smaller than the dimension of the originalfrequency response problem in Equation (1), the solution of the problemtypically becomes more economical.

These modes are obtained as a partial eigensolution of the generalizedeigenvalue problem KΦ=MΦΛ, in which Φ is a rectangular matrix whosecolumns may be eigenvectors, and Λ is a diagonal matrix containingreal-valued eigenvalues, which may be squares of natural frequencies.With mass normalization, so that Φ^(T)MΦ=I, where I is the identitymatrix, Φ and Λ satisfy Φ^(T)KΦ=Λ. By making the approximation X≈ΦY andpre-multiplying the frequency response problem of Equation (1) by Φ^(T),the modal frequency response problem, which is of dimension m, may beobtained in Equation (2) as:{−ω² I+iω(Φ^(T) BΦ)+[(1+iγ)Λ+i(Φ^(T) K ₄Φ)]}Y=Φ ^(T) F.  (2)The accuracy of this modal approximation may be adequate for certainpurposes if the cutoff frequency is chosen appropriately.

The number of modes m represented in Φ can be in the thousands, whichmay reduce the dimension of the original frequency response problem ofEquation (1) to that of the modal frequency response problem of Equation(2). When B and K₄ are not present, the solution of the modal frequencyresponse problem can be less costly since the coefficient matrix mightbe diagonal. However, when B and K₄ are nonzero, the matrices Φ^(T)BΦand Φ^(T)K₄Φ may each be of dimension m×m and fully populated, such thatsolving the modal frequency response problem becomes considerably moreexpensive.

This is because, at every frequency, a complex dense square matrixhaving a dimension in the thousands may have to be factored. The cost offactorization may be proportional to the cube of the matrix dimension,which can be equal to the number of eigenvectors in Φ. As the upperfrequency limit for the analysis increases, so does the cutoff frequencyfor the modes represented in Φ, so the number of modes m may increasemarkedly. Therefore, the expense of the modal frequency responseanalysis can increase rapidly as the upper frequency limit for theanalysis increases. Until recently, frequency response analysis inindustry has generally been limited to low frequencies, lessening thisconcern, mostly because of the cost of the KΦ=MΦΛ eigensolution.

Another approach to the solution of the modal frequency response problemis to diagonalize the coefficient matrix by solving an eigenvalueproblem. Because the coefficient matrix may be quadratic in frequency, astate-space formulation might be used, in which the unknowns includevelocities in addition to displacements. However, this may result indoubling the dimension of the eigenvalue problem, possibly increasingthe cost of the eigensolution by a factor of eight. Complex arithmeticand asymmetry can also add to the cost.

Making use of such an eigensolution approach might be more economical insome cases than the approach of factoring the complex dense coefficientmatrix in Equation (2) at each frequency, but the difference in cost hasnot been enough to motivate implementing the eigensolution approach inmost industrial structural analysis software. Thus, there is a need forapparatus, systems, articles, and methods for more efficientlydetermining the frequency response characteristics of damped structures,including vehicles, such as automobiles, aircraft, ships, submarines,and spacecraft.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow diagram of a method according to various embodiments;and

FIG. 2 illustrates an apparatus, system, and article according tovarious embodiments.

DETAILED DESCRIPTION

To simplify the following discussion, some basic definitions andnotation will be introduced, as follows:

-   M_(R): is a reduced form of a symmetric mass matrix M. However, for    simpler problems, M_(R) may also be treated as an original mass    matrix.-   B_(R): is a reduced form of a viscous damping matrix B. However, for    simpler problems, B_(R) may also be treated as an original viscous    damping matrix.-   K_(R): is a reduced form of a symmetric stiffness matrix K. However,    for simpler problems, K_(R) may also be treated as an original    stiffness matrix.-   γ: is a scalar structural damping parameter.-   K_(4R): is a reduced form of a symmetric structural damping matrix    K₄, which represents local departure from the level of structural    damping represented by γ. However, for simpler problems, K_(4R) may    also be treated as an original structural damping matrix.-   F_(R): is a reduced form of a matrix F including a plurality of load    vectors acting on a structure. Loading may be mechanical, in the    conventional sense, as well as acoustic, fluidic, or otherwise. In    the case of a “structural acoustic” or “fluid-structure interaction”    problem, an acoustic or fluid pressure loading may have to be    determined in the solution process. It is customary to arrange the    equations so that the unknown loading terms are on the left-hand    side of the equations with the unknown structural response, but we    note that these loading terms can equivalently be moved to the    right-hand side of the equations that govern the structural    response, and included in the matrix F. Any type of force that can    be applied to a damped structure can be considered a “load” herein,    and characterized by one or more load vectors included in the    matrix F. Thus, the load vectors may be actual, simulated, or a    combination of these.-   i: is the “imaginary unit,” satisfying i=√(−1).

Considering again Equation (2), a complex symmetric m×m matrix C may bedefined as C=(1+iγ)Λ+i(Φ^(T)K₄Φ). The eigenvalue problem may then bestated as CΦ_(C)=Φ_(C)Λ_(C), for which the two square matrices Φ_(C) anddiagonal Λ_(C) may be complex-valued. Φ_(C) can be normalized to satisfyΦ_(C) ^(T)Φ_(C)=I, in which case Φ_(C) ^(T)CΦ_(C)=Λ_(C). For thefrequency response problem of Equation (2), the matrix Y may be definedas Y=Φ_(C)Z for an arbitrary matrix Y because Φ_(C) may be square andnonsingular. Equation (2) may then be pre-multiplied by Φ_(C) ^(T) toobtain Equation (3):{−ω² I+iω(Φ_(C) ^(T)Φ^(T) BΦΦ _(C))+Λ_(C) }Z=Φ _(C) ^(T)Φ^(T) F.  (3)

Matrix B may have a low rank (e.g., less than about 50, including zero),be real-valued and have a singular value decomposition B=UΣV^(T). Here Uand V may have r columns, where r=rank(B), and Σ may be of dimension r×rand diagonal, containing the nonzero singular values of B. For the casein which B is symmetric, for example, representing viscous dampingeffects, U=V may contain the eigenvectors of B corresponding to nonzeroeigenvalues of B which appear in Σ. Using a singular value decompositionof matrix B, Equation (3) can be transformed to become:(D(ω)+PQ(ω)R)Z=Φ _(C) ^(T)Φ^(T) F.  (4)

As a matter of convenience, it is possible to define a series ofmatrices, such as an m×m diagonal matrix D(ω)=−ω²I+Λ_(C), an m×r matrixP=Φ_(C) ^(T)Φ^(T)U, an r×r diagonal matrix Q(ω)=iωΣ, and an r×m matrixR=V^(T)ΦΦ_(C), wherein U and V are matrices which may satisfy a singularvalue decomposition B=UΣV^(T), and wherein Σ may be a diagonal matrixincluding singular values. The frequency response problem in Equation(4) may then be set forth as:{D(ω)+PQ(ω)R}Z=Φ _(C) ^(T)Φ^(T) F.  (5)

The coefficient matrix (D+PQR) may be referred to as a “diagonal pluslow rank” (DPLR) matrix, the modified frequency response problem(Equation (5)) may be referred to as the DPLR problem, and the solutionmatrix Z may be referred to as the DPLR solution. The inverse of theDPLR matrix is given by Equation (6):(D+PQR)⁻¹ =D ⁻¹ −D ⁻¹ PQ ^(1/2)(I+Q ^(1/2) RD ⁻¹ PQ ^(1/2))⁻¹ Q ^(1/2)RD ⁻¹.  (6)If matrix B is symmetric, matrix R may be equal to P^(T). In this case,the inverse of the DPLR matrix may be transformed to become Equation(7):(D+PQP ^(T))⁻¹ =D ⁻¹ −D ⁻¹ PQ ^(1/2)(I+Q ^(1/2) P ^(T) D ⁻¹ PQ^(1/2))⁻¹(D ⁻¹ PQ ^(1/2))^(T).  (7)Those of ordinary skill in the art, after reading this disclosure, willrealize that there are other equivalent ways of expressing the inverseof the DPLR matrix.

Since matrix B may be of low rank, solving Equation (5) for Z may bevery economical. One method of solving Equation (5) may be to multiplythe right-hand side matrix Φ_(C) ^(T)Φ^(T)F by the inverse of the DPLRmatrix, quite possibly forming the product in stages. Alternatively, aniterative scheme could be used. For example, the DPLR problem (Equation(5)) can be multiplied by D⁻¹ to obtain a modified DPLR problem, setforth in Equation (8):{I+D ⁻¹ PQR}Z=D ⁻¹Φ_(C) ^(T)Φ^(T) F.  (8)An initial approximate solution for Equation (8) might be: Z₀=D⁻¹Φ_(C)^(T)Φ^(T)F. The residual of Equation (8) can be computed and used toform the first block of a block Krylov subspace. The second block of theblock Krylov subspace may be obtained by multiplying the first block byD⁻¹PQR, the third is obtained by multiplying the second block of theblock Krylov subspace by D⁻¹PQR, and so on. If the initial approximatesolution is corrected by minimizing the norm of the residual of themodified DPLR problem (Equation (8)) over the block Krylov subspace,convergence to the exact DPLR solution may occur in r iterations. Once Zis obtained, determining X and Y is straightforward.

If the matrix K₄, in Equation (1) is of relatively low rank (e.g., rankless than about 2000, including zero), forming the matrix C and solvingthe eigenvalue problem CΦ_(C)=Φ_(C)Λ_(C) may become unnecessary. Ifsingular value decompositions of B and K₄ are represented asB=U_(B)Σ_(B)V_(B) ^(T) and K₄=U_(K4)Σ_(K4)V_(K4) ^(T), respectively, themodal frequency response problem of Equation (2) can be written in theform of Equation (9):

$\begin{matrix}{{\left. \left. \left\{ {{{- \omega^{2}}I} + {\left( {1 + {{\mathbb{i}}\;\gamma}} \right)\;\Lambda} + {{{{\Phi^{T}\left\lbrack {U_{B}U_{K4}} \right\rbrack}\;\begin{bmatrix}{{\mathbb{i}}\;\omega\;\Sigma_{B}} & 0 \\0 & {{\mathbb{i}}\;\Sigma_{K4}}\end{bmatrix}}\;\begin{bmatrix}V_{B}^{T} \\V_{K_{4}}^{T}\end{bmatrix}}\Phi}} \right) \right\rbrack \right\}\; Y} = {\Phi^{T}F}} & (9)\end{matrix}$This is in the same form as Equation (5), with D(ω)=−ω²I+(1+iγ)Λ,P=Φ^(T)[U_(B)U_(K4)], Q(ω)=block-diag (iωΣ_(B), iΣ_(K4)), andR=(Φ^(T)[U_(B)U_(K4)])^(T). Z has been replaced by Y, and the right-handside of Equation (9) has become Φ^(T)F. This form of the DPLR problemcan be solved as described above.

In some instances, forming a reduced frequency response problem by usinga reduced subspace does not result in diagonal reduced mass andstiffness matrices (e.g., I and Λ in Equation (2)). This can occur ifdesign modifications result in changes to the finite element mass andstiffness matrices, or if the columns of the matrix Φ, which define thereduced subspace, are not eigenvectors of the eigenvalue problem KΦ=MΦΛ.However, the frequency response approach presented here can still beused. In the case of design modifications which result in matrices M+ΔM(e.g., where ΔM is a change in the mass matrix, B+ΔB, etc., in Equation(1), approximation of the response as X≈ΦY and pre-multiplying thefrequency response problem of Equation (1) by Φ^(T), produces a reducedfrequency response problem, in Equation (10) as:{−ω² M+iωB+[(1+iγ)K+iK ₄ ]}Y=Φ ^(T) F.  (10)in which the reduced matrices are:

-   M=Φ^(T)(M+ΔM)Φ, B=Φ^(T)(B+ΔB)Φ, K=Φ^(T)(K+ΔK)Φ, and    K4=Φ^(T)(K4+ΔK4)Φ.    The reduced problem of Equation (10) can be solved by performing the    factorization    M_(R)=L_(M)L_(M) ^(T), wherein L_(M) is a lower triangular matrix    (note that other types of factorizations, such as one based on an    eigenvalue decomposition of M_(R), could also be used), representing    Y as Y=L_(M) ^(−T)W, and pre-multiplying Equation (10) by L_(M) ⁻¹    to obtain:    {−ω² I+iωL _(M) ⁻¹ BL _(M) ^(−T)+(1+iγ)L _(M) ⁻¹ KL _(M) ^(−T) +iL    _(M) ⁻¹ K ₄L_(M) ^(−T))W=L _(M) ⁻¹Φ^(T) F.  (11)    An intermediate matrix C can then be defined as in Equation (12):    C=(1+iγ)L _(M) ⁻¹ KL _(M) ^(−T) +iL _(M) ⁻¹ K ₄ L _(M) ^(−T).  (12)    The eigenvalue problem CΦ_(C)=Φ_(C)Λ_(C) is solved as described    above, such that Φ_(C) ^(T)Φ_(C)=I, and Φ_(C) ^(T)CΦ_(C)=Λ_(C).    Representing W as W=Φ_(C)Z, and pre-multiplying Equation (11) by    Φ_(C) ^(T) yields Equation (13):    {−ω² I+Λ _(C) +iωΦ _(C) ^(T) L _(M) ⁻¹Φ^(T) UΣV ^(T) ΦL _(M)    ^(−T)Φ_(C))Z=Φ _(C) ^(T) L _(M) ⁻¹Φ^(T) F.  (13)    in which the singular value decomposition B+ΔB=UΣV^(T) has been    used. Equation (13) can be written in the form of Equation (5), with    D(ω) and Q(ω) as before, and P=Φ_(C) ^(T)L_(M) ⁻¹Φ^(T)U and    R=V^(T)ΦL_(M) ^(−T)Φ_(C). Then the right-hand side of Equation (13)    is replaced with Φ_(C) ^(T)L_(M) ⁻¹Φ^(T)F, and the DPLR problem can    be solved using the approach presented previously.

In summary, damped frequency response problems may be solved byaccomplishing (a) some preparatory tasks prior to a sweep through thefrequencies of interest, and then performing (b) other tasks at eachfrequency of interest. The preparatory tasks (a) can be stated as:

-   -   obtaining a matrix whose columns span the subspace used to        approximate the frequency response (typically by solving a        global undamped eigenvalue problem involving the finite element        stiffness and mass matrices),    -   computing a decomposition of a viscous damping matrix, and of        the structural damping matrix if it is of low rank,    -   computing the reduced mass, stiffness, structural damping (if it        is not of low rank), and load matrices which appear in the modal        or reduced frequency response problem,    -   if the structural damping matrix is not of low rank, computing        the eigenvalues and

-   eigenvectors of the matrix which is formed from the modal or reduced    stiffness and

-   structural damping matrices, and    -   computing any other matrices needed for the frequency response        calculations that do        not vary with frequency.

The frequency sweep task (b) may comprise:

-   -   at each frequency, solving the “diagonal plus low rank” (DPLR)        frequency response problem for the DPLR solution by using a        direct or an iterative method.

Building on the principles illustrated above, many embodiments may berealized. For example, FIG. 1 is a flow diagram illustrating a methodaccording to various embodiments. The method 111 may be used to solve anequation associated with a structure, such as a building or anautomobile, of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R). ω may be atime-harmonic excitation frequency, M_(R) may be a reduced form of asymmetric mass matrix M, B_(R) may be a reduced form of a viscousdamping matrix B, and γ may be a scalar global structural dampingcoefficient. K_(R) may be a reduced form of a symmetric stiffness matrixK, K_(4R) may be a reduced form of a symmetric structural damping matrixK₄ representing local departures from γ, and F_(R) may be a reduced formof a matrix F including a plurality of load vectors acting on thestructure.

The method 111 may comprise factoring the reduced matrixM_(R)=L_(M)L_(M) ^(T), wherein L_(M) is a lower triangular matrix (notethat other types of factorizations, such as one based on an eigenvaluedecomposition of M_(R), could also be used) at block 121 and defining anintermediate matrix C=(1+iγ)L_(M) ⁻¹K_(R)L_(M) ^(−T)+iL_(M)⁻¹K_(4R)L_(M) ^(−T); at block 125. The method 111 may also includecomputing a plurality of eigenvalues and eigenvectors for a secondequation CΦ_(C)=Φ_(C)Λ_(C), wherein Φ_(C) includes eigenvectors of amatrix C and Φ_(C) is normalized so that Φ_(C) ^(T)Φ_(C)=I, at block131. I may be an identity matrix, and Λ_(C) may be diagonal and includeeigenvalues of the matrix C. Alternatively, the generalized eigenvalueproblem [(1+iγ)K_(R)+iK_(4R)]Φ_(G)=M_(R)Φ_(G)Λ_(C), wherein Φ_(G)=L_(M)^(−T)Φ_(C), can be solved directly, without factoring the reduced matrixM_(R).

The method 111 may further include computing a matrix P=Φ_(C) ^(T)L_(M)⁻¹U_(R) and a matrix R=V_(R) ^(T)L_(M) ^(−T)Φ_(C), wherein U_(R) andV_(R) are matrices which may satisfy a singular value decompositionB_(R)=U_(R)Σ_(R)V_(R) ^(T), at block 135. Σ_(R) may be diagonal,conformal with U_(R) and V_(R), and include singular values. The method111 may also include solving an equation of the form(D(ω)+PQ(ω)R)Z=Φ_(C) ^(T)L_(M) ⁻¹F_(R) for Z at block 151 for eachfrequency of interest, wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ_(R). Themethod may conclude at block 155 with forming a product Y=L_(M)^(−T)Φ_(C)Z, wherein the product Y may be a matrix having a plurality ofvectors corresponding to the plurality of load vectors (acting on thestructure) included in the matrix F.

Many variations of the method 111 are possible. For example, the method111 may include approximately solving an equation of the form{−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the equation by a matrixΦ^(T) and solving for the matrix Y at block 155. The matrix X may beapproximately equal to ΦY, and the matrix Φ may be a rectangular matrixhaving columns that span the approximating subspace. The method 111 maycontinue with computing the reduced matrices M_(R)=Φ^(T)MΦ,B_(R)=Φ^(T)BΦ, K_(R)=Φ^(T)KΦ, K_(4R)=Φ^(T)K₄Φ, and F_(R)=Φ^(T)F, andthen forming an approximation X≈ΦY at block 155.

The method 111 may also include providing the matrix Φ, as well asproviding or obtaining the matrices M, B, K, K₄, and F, and the scalarγ, which make take the form of experimental data. As noted previously,the method 111 may also comprise solving an equation of the form(D(ω)+PQ(ω)R)Z=Φ_(C) ^(T)L_(M) ⁻¹F_(R) for Z directly or iteratively, aswell as repeatedly solving for Z for each one of a selected set ofexcitation frequencies including the excitation frequency ω at block151.

Other variations of the method 111 are possible. For example, in theinstance where matrices M and K are diagonalized by Φ, the method 111may be used for solving an equation, perhaps associated with astructure, of the form{−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F. ω may be atime-harmonic excitation frequency, I₁ may be an identity matrix whichsatisfies an equation of the form Φ^(T)MΦ=I₁, and M may be a symmetricmass matrix. Φ may be a matrix having a plurality of columns, eachcolumn including an eigenvector approximating one of a correspondingplurality of undamped natural modes of vibration associated with thestructure, and B may be a viscous damping matrix. γ may be a scalarrepresenting a global structural damping coefficient, and Λ may be adiagonal matrix which satisfies an equation of the form Φ^(T)KΦ=Λ,wherein K is a symmetric stiffness matrix. K₄ may be a symmetricstructural damping matrix representing local departures from the globalstructural damping coefficient γ, and Y may be a matrix from which theapproximation X≈ΦY is made, wherein X is a matrix having a plurality ofdisplacement vectors to be determined and corresponding to a pluralityof load vectors included in the matrix F.

In these circumstances, the method 111 may comprise defining anintermediate matrix C=(1+iγ)Λ+i(Φ^(T)K₄Φ) at block 125 and computing aplurality of eigenvalues and eigenvectors for a third equationCΦ_(C)=Φ_(C)Λ_(C), at block 131. Φ_(C) may include eigenvectors of thematrix C, such that Φ_(C) is normalized so that Φ_(C) ^(T)Φ_(C)=I₂, asecond identity matrix. Λ_(C) may be a diagonal matrix containingeigenvalues of C.

The method 111 may also include computing a matrix P=Φ_(C) ^(T)Φ^(T)Uand a matrix R=V^(T)ΦΦ_(C) at block 135. U and V may be matrices whichsatisfy a singular value decomposition B=UΣV^(T), and Σ may be adiagonal matrix including singular values. The method 111 may alsoinclude solving an equation of the form {D(ω)+PQ(ω)R}Z=Φ_(C) ^(T)Φ^(T)Ffor Z at block 151, wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ, and thencomputing Y=Φ_(C) ^(T)Z at block 155.

In this case also, the method 111 provides approximation alternatives,such that the method 111 may include approximately solving an equationof the form {−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the equationby Φ^(T) and solving for a matrix Y, wherein the matrix X isapproximately equal to ΦY, wherein the matrix Φ is a matrix satisfyingthe equations KΦ=MΦΛ, Φ^(T)MΦ=I₃, a third identity matrix, andΦ^(T)KΦ=Λ. The matrix Φ may include a plurality of columns, each columnincluding an eigenvector that approximates one of a correspondingplurality of undamped natural modes of vibration associated with thestructure. The matrix Λ may be a diagonal matrix including a pluralityof eigenvalues, each one of the plurality of eigenvalues correspondingto one of the eigenvectors or columns included in the matrix Φ. Thematrix Y may include a plurality of vectors corresponding to theplurality of load vectors included in the matrix F. And, as noted above,the method 111 may include forming an approximation X≈ΦY.

In this case, the method 111 may also include computing the matrix Φ, aswell as providing the matrices M, B, K, K₄, and F, and the scalar γ,which may include the acquisition of experimental data with respect tothe structure. In addition, the equation of the form{D(ω)+PQ(ω)R}Z=Φ_(C) ^(T)Φ^(T)F may be solved for Z directly, oriteratively, and include repeatedly solving for Z using each one of aselected set of excitation frequencies including the excitationfrequency ω.

Other variations of the method 111 are possible. For example, in theinstance where matrices M and K are diagonalized by Φ, a singular valuedecomposition of B_(R)=Φ^(T)BΦ may be formed. In this case, the method111 may be applied to solving an equation associated with a structure ofthe form {−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F. ω may be atime-harmonic excitation frequency, I₁ may be an identity matrix whichsatisfies an equation of the form Φ^(T)MΦ=I₁ (wherein M is a symmetricmass matrix), and Φ may be a matrix having a plurality of columns, eachcolumn including an eigenvector that approximates one of a correspondingplurality of undamped natural modes of vibration associated with thestructure satisfying the eigenvalue problem KΦ=MΦΛ. B may be a viscousdamping matrix, γ may be a scalar representing a global structuraldamping coefficient, and Λ may be a diagonal matrix which satisfies anequation of the form Φ^(T)KΦ=Λ, wherein K is a symmetric stiffnessmatrix. K₄ may be a symmetric structural damping matrix representinglocal departures from the global structural damping coefficient γ, and Ymay be a matrix from which the approximation X≈ΦY is made, wherein X isa matrix having a plurality of displacement vectors corresponding to aplurality of load vectors acting on the structure and included in thematrix F.

The method 111 in this case may include defining an intermediate matrixC=(1+iγ)Λ+i(Φ^(T)K₄Φ) at block 125 and computing a plurality ofeigenvalues and eigenvectors at block 131 for an equation of the formCΦ_(C)=Φ_(C)Λ_(C), wherein Φ_(C) includes eigenvectors of the matrix C,and Φ_(C) is normalized so that Φ_(C) ^(T)Φ_(C)=I₂, a second identitymatrix. Λ_(C) may be a diagonal matrix containing the eigenvalues of C.

The method 111 may also include computing a matrix P=Φ_(C) ^(T)U_(R),and a matrix R=V_(R) ^(T)Φ_(C) at block 135. U_(R) and V_(R) may bematrices that satisfy a singular value decompositionΦ^(T)BΦ=U_(R)Σ_(R)V_(R) ^(T), and Σ_(R) may be a diagonal matrixincluding singular values. The method 111 may continue with solving anequation of the form {D(ω)+PQ(ω)R}Z=Φ_(C) ^(T)Φ^(T)F for Z at block 151,wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ_(R), and then conclude at block 155with computing Y=Φ_(C) ^(T)Z.

In this case also, the method 111 provides approximation alternatives,such that the method 111 may include approximately solving an equationof the form {−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the equationby Φ^(T) and solving for a matrix Y, wherein the matrix X isapproximately equal to ΦY. The matrix Φ may be a matrix satisfying theequations KΦ=MΦΛ, Φ^(T)MΦ=I₃, a third identity matrix, and Φ^(T)KΦ=Λ,wherein the matrix Φ includes a plurality of columns, each columnincluding an eigenvector that approximates one of a correspondingplurality of natural modes of vibration associated with the structure.The matrix Y may be a matrix having a plurality of vectors correspondingto the plurality of load vectors included in the matrix F. The methodmay then include forming an approximation X=ΦY at block 155. As notedabove, the method 111 may include computing the matrix Φ, as well asproviding the matrices M, B, K, K₄, and F, and the scalar γ, which mayinclude acquiring various data associated with the structure. Inaddition, the method 111 may include solving an equation of the form{D(ω)+PQ(ω)R}Z=Φ_(C) ^(T)Φ^(T)F for Z directly or iteratively, as wellas repeatedly solving for Z using each one of a selected set ofexcitation frequencies including the excitation frequency ω.

Other variations of the method 111 are possible. For example, in theinstance where matrix K₄ is of relatively low rank (e.g., the rank ofmatrix K₄ is less than about 2000, including zero), the method 111 maybe applied to solving an equation associated with a structure of theform {−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F. ω may be atime-harmonic excitation frequency, I₁ may be an identity matrix whichsatisfies an equation of the form Φ^(T)MΦ=I₁, and M may be a symmetricmass matrix. Φ may be a matrix having columns spanning a subspace usedto approximate a plurality of undamped natural modes of vibrationassociated with the structure, and B may be a viscous damping matrix. γmay be a scalar representing a global structural damping coefficient,and Λ may be a matrix that satisfies an equation of the form Φ^(T)KΦ=Λ,wherein K is a symmetric stiffness matrix. K₄ may be a symmetricstructural damping matrix representing local departures from the globalstructural damping coefficient, and Y may be a matrix from which theapproximation X≈ΦY is made, wherein X is a matrix having a plurality ofdisplacement vectors corresponding to a plurality of load vectorsapplied to the structure and included in the matrix F.

In this case, the method 111 may comprise computing a singular valuedecomposition of at least one of the matrices B and K₄, whereinB=U_(B)Σ_(B)V_(B) ^(T), and wherein K₄=U_(K4)Σ_(K4)V_(K4) ^(T); as partof computing matrices P and R at block 135, and then generating matricesp=Φ^(T)[U_(B)U_(K4)] and R=(Φ^(T)[V_(B)V_(K4)])^(T). In this case it canbe seen that D(ω)=−ω²I+(1+iγ)Λ and Q(ω)=block-diag (iωΣ_(B), iΣ_(K4))or:

$Q = {\begin{bmatrix}{{\mathbb{i}}\;\omega\;\Sigma_{B}} & 0 \\0 & {{\mathbb{i}}\;\Sigma_{K4}}\end{bmatrix}.}$The method 111 may include solving an equation of the form{D(ω)+PQ(ω)R}Y=Φ^(T)F for Y.

In this case also, the method 111 provides approximation alternatives,such that the method 111 may include approximately solving an equationof the form {−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the equationby Φ^(T) and solving for a matrix Y. Matrix X may be approximately equalto ΦY, and the matrix Φ may be a matrix satisfying the equations KΦ=MΦΛ,Φ^(T)MΦ=I₂, a second identity matrix, and Φ^(T)KΦ=Λ, wherein the matrixΦ includes a plurality of columns, each column including an eigenvectorapproximating one of a corresponding plurality of natural modes ofvibration associated with the structure. The matrix Y may be a matrixhaving a plurality of vectors corresponding to the plurality of loadvectors included in the matrix F. The method 111 may then includeforming an approximation X≈ΦY at block 155, as well as computing thematrix Φ, as well as providing the matrices M, B, K, K₄, and F, and thescalar γ, to include acquiring data associated with the structure. Anequation of the form {D(ω)+PQ(ω)R}Y=Φ^(T)F may be solved for Y directly,or iteratively, and the method 111 includes repeatedly solving for Yusing each one of a selected set of excitation frequencies (includingthe excitation frequency ω).

In this case, the method 111 may also comprise computing a singularvalue decomposition for each one of the reduced matrices B_(R) andK_(4R), to solve an equation associated with a structure of the form{−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F. Again, ω may be atime-harmonic excitation frequency, I₁ may be an identity matrix whichsatisfies an equation of the form Φ^(T)MΦ=I₁, and M may be a symmetricmass matrix, wherein Φ is a matrix having columns spanning a subspaceused to approximate a plurality of undamped natural modes of vibrationassociated with the structure. The matrix B may be a viscous dampingmatrix, γ may be a scalar representing a global structural dampingcoefficient, and Λ may be a matrix which satisfies a second equation ofthe form Φ^(T)KΦ=Λ, wherein K is a symmetric stiffness matrix. Matrix K₄may be a symmetric structural damping matrix representing localdepartures from the global structural damping coefficient, and Y may bea matrix from which the approximation X≈ΦY is made, wherein X may be amatrix having a plurality of displacement vectors corresponding to aplurality of load vectors included in the matrix F.

In this case, the method 111 may comprise computing a singular valuedecomposition for at least one of the matrices B_(R)=Φ_(T)BΦ andK_(4R)=Φ^(T)K₄Φ, wherein B_(R)=U_(BR)Σ_(BR)V_(BR) ^(T), and whereinK_(4R)=U_(K4R)Σ_(K4R)V_(K4R) ^(T), and then forming a matrixP=[U_(BR)U_(K4R)] and a matrix R=[V_(BR)V^(K4R)]^(T) at block 135. Themethod 111 may continue with solving an equation of the form{D(ω)+PQ(ω)R}Y=Φ^(T)F for Y, wherein D(ω)=−ω²I+(1+iγ)Λ andQ(ω)=block-diag (iωΣ_(B), iΣ_(K4)), as described previously.

In this case also, the method 111 provides approximation alternatives,such that the method 111 may include approximately solving an equationof the form {−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the equationby Φ^(T) and solving for a matrix Y, wherein the matrix X isapproximately equal to ΦY. The matrix Φ may be a matrix satisfying theequations KΦ=MΦΛ, Φ^(T)MΦ=I₂, a second identity matrix, and Φ^(T)KΦ=Λ,wherein the matrix Φ includes a plurality of columns, each columnincluding an eigenvector that approximates one of a correspondingplurality of natural modes of vibration associated with the structure.The matrix Y may be a matrix having a plurality of vectors correspondingto the plurality of load vectors included in the matrix F. In addition,the method may include forming an approximation X≈ΦY at block 155, aswell as computing the matrix Φ, and providing the matrices M, B, K, K₄,and F, and the scalar γ (which may include the acquisition of dataassociated with the structure). The equation of the form{D(ω)+PQ(ω)R}Y=Φ^(T)F may be solved for Y directly, or iteratively, aswell as repeatedly solving for Y using each one of a selected set ofexcitation frequencies including the excitation frequency ω.

In summary, many variations of the illustrated embodiments may berealized. Immediately following are a few of the possible variations.

A method of solving a first equation of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), and associated with astructure may be realized. ω may be a time-harmonic excitationfrequency, M_(R) may be a reduced form of a symmetric mass matrix M,B_(R) may be a reduced form of a viscous damping matrix B, γ may be ascalar global structural damping coefficient, K_(R) may be a reducedform of a symmetric stiffness matrix K, K_(4R) may be a reduced form ofa symmetric structural damping matrix K₄ representing local departuresfrom γ, and F_(R) may be a reduced form of a matrix F including aplurality of load vectors acting on the structure. The method maycomprise solving an eigenvalue problem of the form[(1+iγ)K_(R)+iK_(4R)]Φ_(G)=M_(R)Φ_(G)Λ_(G); computing a matrix P=Φ_(G)^(T)U_(R) and a matrix R=V_(R) ^(T)Φ_(G), wherein U_(R) and V_(R) arematrices which satisfy a singular value decompositionB_(R)=U_(R)Σ_(R)V_(R) ^(T), and wherein Σ_(R) is diagonal, conformalwith U_(R) and V_(R), and includes singular values; solving a thirdequation of the form (D(ω)+PQ(ω)R)Z=Φ_(G) ^(T)F_(R) for Z, whereinD(ω)=−ω²I+Λ_(G) and Q(ω)=iωΣ_(R), wherein Φ_(G) is normalized to satisfyΦ_(G) ^(T)M_(R)Φ_(G)=I, where I is an identity matrix and Φ_(G)^(T)K_(R)Φ_(G=Λ) _(G); and forming a product Y=L_(M) ^(−T)Φ_(G)Z,wherein the product Y is a matrix having a plurality of vectorscorresponding to the plurality of load vectors included in the matrix F.

Another method of solving a first equation of the form{−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, and associated with a structure may berealized. Here X may be a matrix of displacement vectors, ω may be atime-harmonic excitation frequency, M may be a symmetric mass matrix, Bmay be a viscous damping matrix, γ may be a scalar global structuraldamping coefficient, K may be a symmetric stiffness matrix, K₄ may be asymmetric structural damping matrix representing local departures fromγ, F may be a matrix including a plurality of load vectors acting on thestructure. In this case, the method may comprise transforming at leasttwo matrices included in a set of matrices comprising M, B, K, K₄ toprovide a set of at least two diagonalized matrices and a set ofnon-diagonalized matrices, wherein K₄ is non-null; forming a matrix Dcomprising a linear combination of the set of at least two diagonalizedmatrices; forming a matrix product PQR as a representation of a linearcombination of the set of non-diagonalized matrices; solving a secondequation of the form (D+PQR)Z=A for Z, wherein Z is a frequency responsesolution matrix, and wherein A is a transformed load matrix; andback-transforming the frequency response solution matrix Z to providethe matrix X. The matrices P and R can be rectangular, and the matrix Qcan be square.

Yet another method of solving a first equation of the form{−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, and associated with a structure may berealized. Here X may be a matrix of displacement vectors, ω may be atime-harmonic excitation frequency, M may be a symmetric mass matrix, Bmay be a viscous damping matrix, γ may be a scalar global structuraldamping coefficient, K may be a symmetric stiffness matrix, K₄ may be asymmetric structural damping matrix representing local departures fromγ, F may be a matrix including a plurality of load vectors acting on thestructure. In this case, the method may comprise forming a matrix Dcomprising a linear combination of a set of at least two diagonalmatrices selected from the set of matrices M, B, K, K₄, wherein at leasttwo of the matrices M, B, K, K₄ are diagonal, and wherein at least oneof the matrices B and K₄ is non-null; forming a matrix product PQR as arepresentation of a linear combination of a set of non-diagonal matricesselected from the set of matrices M, B, K, K₄; and solving a secondequation of the form (D+PQR)X=F for the matrix X. The matrices P and Rcan be rectangular, and the matrix Q can be square.

In another variation, a method of approximating a solution of a firstequation associated with a structure, the first equation being of theform {−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, and associated with a structure maybe realized. Here X may be a matrix of displacement vectors, ω may be atime-harmonic excitation frequency, M may be a symmetric mass matrix, Bmay be a viscous damping matrix, γ may be a scalar global structuraldamping coefficient, K may be a symmetric stiffness matrix, K₄ may be asymmetric structural damping matrix representing local departures fromγ, F may be a matrix including a plurality of load vectors acting on thestructure. In this case, the method may comprise obtaining reduced formsof the matrices M, B, K, K₄, and F as M_(R)=Φ^(T)MΦ, B_(R)=Φ^(T)BΦ,K_(R)=Φ^(T)KΦ, K_(4R)=Φ^(T)K₄Φ, and F_(R)=Φ^(T)FΦ, respectively, whereinΦ is a matrix having columns spanning a subspace of approximation forapproximating the solution matrix X, and wherein at least one of thematrices B and K₄ is non-null; forming a matrix D comprising a linearcombination of a set of at least two diagonalized matrices selected froma set of matrices obtained from transforming the reduced matrices M_(R),B_(R), K_(R), and K_(4R); forming a matrix product PQR as arepresentation of a linear combination of the set of non-diagonalizedmatrices selected from a set of matrices obtained from transforming thereduced matrices M_(R), B_(R), K_(R), and K_(4R); solving a secondequation of the form (D+PQR)Z=A for Z, wherein Z is a frequency responsesolution matrix, and wherein A is a transformed load matrix; andback-transforming the frequency response solution matrix Z to provide aback-transformed matrix Z and multiplying the back-transformed matrix Zby Φ to provide an approximation of the solution matrix X. The matricesP and R can be rectangular, and the matrix Q can be square.

Finally, in yet another variation, a method of approximating a solutionof a first equation associated with a structure, the first equationbeing of the form {−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, and associated with astructure may be realized. Here X may be a matrix of displacementvectors, ω may be a time-harmonic excitation frequency, M may be asymmetric mass matrix, B may be a viscous damping matrix, γ may be ascalar global structural damping coefficient, K may be a symmetricstiffness matrix, K₄ may be a symmetric structural damping matrixrepresenting local departures from γ, F may be a matrix including aplurality of load vectors acting on the structure. In this case, themethod may comprise obtaining reduced forms of the matrices M, B, K, K₄,and F as M_(R)=Φ^(T)MΦ, B_(R)=Φ^(T)BΦ, K_(R)=Φ^(T)KΦ, K_(4R)=Φ^(T)K₄Φ,and F_(R)=Φ^(T)FΦ, respectively, wherein Φ is a matrix having columnsspanning a subspace of approximation for approximating the solutionmatrix X, wherein at least one of the matrices B and K₄ is non-null, andwherein at least two of the matrices selected from a set of reducedmatrices including M_(R), B_(R), K_(R), and K_(4R) are diagonal; forminga matrix D comprising a linear combination of the at least two diagonalmatrices selected from the set of reduced matrices including thematrices M_(R), B_(R), K_(R), and K_(4R); forming a matrix product PQRas a representation of a linear combination of the set of non-diagonalmatrices selected from a set of reduced matrices including the matricesM_(R), B_(R), K_(R), and K_(4R); solving a second equation of the form(D+PQR)Z=F_(R) for Z, wherein Z is a frequency response solution matrix;and multiplying the matrix Z by the matrix Φ to provide an approximationof the solution matrix X. Again, the matrices P and R can berectangular, and the matrix Q can be square.

It should be noted that the methods described herein do not have to beexecuted in the order described, or in any particular order. Moreover,various activities described with respect to the methods identifiedherein can be executed in serial or parallel fashion. For the purposesof this document, the terms “information” and “data” may be usedinterchangeably. Information, including parameters, commands, operands,and other data, can be sent and received in the form of one or morecarrier waves.

Upon reading and comprehending the content of this disclosure, one ofordinary skill in the art will understand the manner in which a softwareprogram can be launched from a computer readable medium in acomputer-based system to execute the functions defined in the softwareprogram. One of ordinary skill in the art will further understand thevarious programming languages that may be employed to create one or moresoftware programs designed to implement and perform the methodsdisclosed herein. The programs may be structured in an object-orientedformat using an object-oriented language such as Java, Smalltalk, orC++. Alternatively, the programs can be structured in aprocedure-orientated format using a procedural language, such asassembly or C. The software components may communicate using any of anumber of mechanisms well-known to those skilled in the art, such asapplication program interfaces or interprocess communication techniques,including remote procedure calls. The teachings of various embodimentsof the present invention are not limited to any particular programminglanguage or environment, including Hypertext Markup Language (HTML) andExtensible Markup Language (XML).

Thus, other embodiments may be realized. FIG. 2 illustrates an apparatus222, system 220, and article 220 according to various embodiments. Forexample, an article 220 according to various embodiments, such as acomputer, a memory system, a magnetic or optical disk, some otherstorage device, and/or any type of electronic device or system maycomprise a machine-accessible medium 228 such as a memory (e.g., amemory including an electrical, optical, or electromagnetic conductor)having associated data 244, 248, 252, 256, 260, and 264 (e.g., computerprogram instructions), which when accessed, results in a machine 224performing such actions as solving an equation associated with astructure. The equation may be of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), wherein ω is atime-harmonic excitation frequency, M_(R) is a reduced form of asymmetric mass matrix M, B_(R) is a reduced form of a viscous dampingmatrix B, γ is a scalar global structural damping coefficient, K_(R) isa reduced form of a symmetric stiffness matrix K, K_(4R) is a reducedform of a symmetric structural damping matrix K₄ representing localdepartures from γ, and F_(R) is a reduced form of a matrix F including aplurality of load vectors. The data, when accessed, may result in amachine performing such actions as factoring the reduced matrixM_(R)=L_(M)L_(M) ^(T), wherein L_(M) is a lower triangular matrix (notethat other types of factorizations, such as one based on an eigenvaluedecomposition of M_(R), could also be used), defining an intermediatematrix C=(1+iγ)L_(M) ⁻¹K_(R)L_(M) ^(−T)+iL_(M) ⁻¹K_(4R)L_(M) ^(−T),computing a plurality of eigenvalues and eigenvectors for a secondequation CΦ_(C)=Φ_(C)Λ_(C), wherein Φ_(C) includes eigenvectors of amatrix C and Φ_(C) is normalized so that Φ_(C) ^(T)Φ_(C)=I (I may be anidentity matrix and Λ_(C) may be diagonal and include eigenvalues of thematrix C). Alternatively, the generalized eigenvalue problem[(1+iγ)K_(R)+iK_(4R)]Φ_(G)=M_(R)Φ_(G)Λ_(C), wherein Φ_(G)=L_(M)^(−T)Φ_(C), can be solved directly, without factoring the reduced matrixM_(R).

Other actions performed by the machine 224 may include computing amatrix P=Φ_(C) ^(T)L_(M) ⁻¹U_(R) and a matrix R=V_(R) ^(T)L_(M)^(−T)Φ_(C), wherein U_(R) and V_(R) are matrices which satisfy asingular value decomposition B_(R)=U_(R)Σ_(R)V_(R) ^(T), and whereinΣ_(R) is diagonal, conformal with U_(R) and V_(R), and includes singularvalues; and solving an equation of the form (D(ω)+PQ(ω)R)Z=Φ_(C)^(T)L_(M) ⁻¹F_(R) for Z, wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ_(R).Further actions performed by the machine 224 may include forming aproduct Y=L_(M) ^(−T)Φ_(C)Z, wherein the product Y is a matrix having aplurality of vectors corresponding to the plurality of load vectorsincluded in the matrix F.

Still other embodiments may be realized. For example, some embodimentsmay include an apparatus to solve an equation associated with astructure. The equation may have the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), wherein ω is atime-harmonic excitation frequency, M_(R) is a reduced form of asymmetric mass matrix M, B_(R) is a reduced form of a viscous dampingmatrix B, and γ is a scalar global structural damping coefficient. K_(R)may be a reduced form of a symmetric stiffness matrix K, and K_(4R) maybe a reduced form of a symmetric structural damping matrix K₄representing local departures from γ. F_(R) may be a reduced form of amatrix F including a plurality of load vectors acting on the structure238.

The apparatus 222 may comprise a factoring module 244 to factor thereduced matrix M_(R)=L_(M)L_(M) ^(T), wherein L_(M) is a lowertriangular matrix (note that other types of factorizations, such as onebased on an eigenvalue decomposition of M_(R), could also be used), anda definition module 248 capable of being communicatively coupled to thefactoring module. The definition module 248 is to define an intermediatematrix C=(1+iγ)L_(M) ⁻¹K_(R)L_(M) ^(−T)+iL_(M) ⁻¹K_(4R)L_(M) ^(−T).

The apparatus 222 may also include a first computation module 252capable of being communicatively coupled to the definition module. Thefirst computation module 252 is to compute a plurality of eigenvaluesand eigenvectors for an equation of the form CΦ_(C)=Φ_(C)Λ_(C), whereinΦ_(C) includes eigenvectors of a matrix C and Φ_(C) is normalized sothat Φ_(C) ^(T)Φ_(C)=I, wherein I is an identity matrix, and whereinΛ_(C) is diagonal and includes eigenvalues of the matrix C(alternatively, the generalized eigenvalue problem[(1+iγ)K_(R)+iK_(4R)]Φ_(G)=M_(R)Φ_(G)Λ_(C), wherein Φ_(G)=L_(M)^(−T)Φ_(C), can be solved directly, without factoring the reduced matrixM_(R)). The apparatus 222 may also include a second computation module256 capable of being communicatively coupled to the first computationmodule 252. The second computation module 256 is to compute a matrixP=Φ_(C) ^(T)L_(M) ⁻¹U_(R) and a matrix R=V_(R) ^(T)L_(M) ^(−T)Φ_(C),wherein U_(R) and V_(R) are matrices that may satisfy a singular valuedecomposition B_(R)=U_(R)Σ_(R)V_(R) ^(T). Σ_(R) may be diagonal,conformal with U_(R) and V_(R), and includes singular values.

The apparatus 222 may include an equation solving module 260 capable ofbeing communicatively coupled to the second computation module 256. Theequation solving module 260 is to solve an equation of the form(D(ω)+PQ(ω)R)Z=Φ_(C) ^(T)L_(M) ⁻¹F_(R) for Z, wherein D(ω)=−ω²I+Λ_(C)and Q(ω)=iωΣ_(R). The apparatus 222 may also include a product formationmodule 264 to form a product Y=L_(M) ^(−T)Φ_(C)Z, wherein the product Ymay be a matrix having a plurality of vectors corresponding to theplurality of load vectors included in the matrix F.

A system 220 to solve an equation associated with a structure 238 isalso included in some embodiments. The equation may be of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), and include thevariables described in conjunction with the apparatus 222. The system220 may include a processor 224, as well as the apparatus 222 describedabove, perhaps comprising a memory 228 capable of being communicativelycoupled to the processor. The apparatus 222, as described above, mayinclude the modules 244, 248, 252, 256, 260, and 264. As part of theapparatus 222 and system 220 (e.g., a computer), a user interface (I/F)230, perhaps comprising a keyboard, display, and graphical userinterface, may be communicatively coupled to the machine 224 (e.g.,comprising a CPU (central processing unit)), and/or themachine-accessible medium 224 (e.g., comprising a memory).

The apparatus 222 and system 220 have component elements which can allbe characterized as “modules” herein. Such modules can include hardware,circuitry, and/or a microprocessor and/or memory circuits, softwareprogram modules, and/or firmware, and combinations thereof, as desiredby the architect of the system and apparatus, and appropriate forparticular embodiments.

The system 220 may comprise a computer, digital signal processor, orhybrid (digital/analog) computer. The system 220 may be coupled to anetwork adapter 234 and then to a structure 238, or directly to thestructure 238 that is to be characterized (in terms of vibration) by theapparatus, articles, methods, and systems disclosed herein. Coupling maybe electric, mechanical, fluid, thermodynamic, or a combination ofthese.

Implementing the apparatus, systems, and methods described herein mayresult in reducing the time required to solve equations of the generalform {−ω²I+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i((Φ^(T)K₄Φ)]}Y=Φ^(T)F. Such equationsmay be associated with structures undergoing vibration analysis. Ifcertain circumstances exist, for example, that matrix K₄ is ofrelatively low rank (e.g., matrix K₄ has a rank of less than about 2000,including zero), even greater savings may result.

The accompanying drawings that form a part hereof, show by way ofillustration, and not of limitation, specific embodiments in which thesubject matter may be practiced. The embodiments illustrated aredescribed in sufficient detail to enable those skilled in the art topractice the teachings disclosed herein. Other embodiments may beutilized and derived therefrom, such that structural and logicalsubstitutions and changes may be made without departing from the scopeof this disclosure. This Detailed Description, therefore, is not to betaken in a limiting sense, and the scope of various embodiments isdefined only by the appended claims, along with the full range ofequivalents to which such claims are entitled.

Thus, although specific embodiments have been illustrated and describedherein, it should be appreciated that any arrangement calculated toachieve the same purpose may be substituted for the specific embodimentsshown. This disclosure is intended to cover any and all adaptations orvariations of various embodiments. Combinations of the aboveembodiments, and other embodiments not specifically described herein,will be apparent to those of skill in the art upon reviewing the abovedescription.

The Abstract of the Disclosure is provided to comply with 37 C.F.R.§1.72(b), requiring an abstract that will allow the reader to quicklyascertain the nature of the technical disclosure. It is submitted withthe understanding that it will not be used to interpret or limit thescope or meaning of the claims. In addition, in the foregoing DetailedDescription, it can be seen that various features are grouped togetherin a single embodiment for the purpose of streamlining the disclosure.This method of disclosure is not to be interpreted as reflecting anintention that the claimed embodiments require more features than areexpressly recited in each claim. Rather, as the following claimsreflect, inventive subject matter lies in less than all features of asingle disclosed embodiment. Thus the following claims are herebyincorporated into the Detailed Description, with each claim standing onits own as a separate embodiment.

1. A method of solving a first equation associated with a structure, thefirst equation being of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), wherein ω is atime-harmonic excitation frequency, wherein M_(R) is a reduced form of asymmetric mass matrix M, wherein B_(R) is a reduced form of a viscousdamping matrix B, wherein γ is a scalar global structural dampingcoefficient, wherein K_(R) is a reduced form of a symmetric stiffnessmatrix K, wherein K_(4R) is a reduced form of a symmetric structuraldamping matrix K₄ representing local departures from γ, and whereinF_(R) is a reduced form of a matrix F including a plurality of loadvectors acting on the structure, the method comprising: factoring thereduced matrix M_(R)=L_(M)L_(M) ^(T), wherein L_(M) is a lowertriangular matrix; defining an intermediate matrix C=(1+iγ)L_(M)⁻¹K_(R)L_(M) ^(−T)+iL_(M) ⁻¹K_(4R)L_(M) ^(−T); computing a plurality ofeigenvalues and eigenvectors for a second equation CΦ_(C)=Φ_(C)Λ_(C),wherein Φ_(C) is a matrix including eigenvectors of the matrix C, andwherein Λ_(C) is a matrix including eigenvalues of the matrix C;computing a matrix P=Φ_(C) ^(T)L_(M) ⁻¹U_(R) and a matrix R=V_(R)^(T)L_(M) ^(−T)Φ_(C), wherein U_(R) and V_(R) are matrices which satisfya singular value decomposition B_(R)=U_(R)Σ_(R)V_(R) ^(T), and whereinΣ_(R) is diagonal, conformal with U_(R) and V_(R), and includes singularvalues; solving a third equation of the form (D(ω)+PQ(ω)R)Z=Φ_(C)^(T)L_(M) ⁻¹F_(R) for Z, wherein D(ω)=−ω²Φ_(C) ^(T)Φ_(C)+Φ_(C)^(T)CΦ_(C) and Q(ω)=iωΣ_(R); forming a product Y=L_(M) ^(−T)Φ_(C)Z,wherein Φ comprises eigenvectors satisfying the eigenvalue problemKΦ=MΦΛ, wherein Λ=Φ^(T)KΦ, and wherein a matrix Φ multiplied by theproduct Y is approximately equal to a matrix X including a plurality ofdisplacements of the structure; and communicating an approximation of avibration-induced displacement of the structure, including at least aportion of a product comprising the matrix Φ multiplied by the productY, to a user interface.
 2. The method of claim 1, further comprising:approximately solving a fourth equation of the form{−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the fourth equation by amatrix Φ^(T) and solving for the matrix Y, wherein the matrix X isapproximately equal to ΦY, and wherein the matrix Φ is a rectangularmatrix having columns that span the approximating subspace; computingthe reduced matrices M_(R)=Φ^(T)MΦ, B_(R)=Φ^(T)BΦ, K_(R)=Φ^(T)KΦ,K_(4R)=Φ^(T)K₄Φ, and F_(R)=Φ^(T)F; and forming an approximation X≈ΦY. 3.The method of claim 2, further comprising: providing the matrix Φ. 4.The method of claim 3, further comprising: providing the matrices M, B,K, K₄, and F, and the scalar γ.
 5. The method of claim 1, furthercomprising: solving the third equation of the form (D(ω)+PQ(ω)R)Z=Φ_(C)^(T)L_(M) ⁻¹F_(R) for Z directly.
 6. The method of claim 1, furthercomprising: solving the third equation of the form (D(ω)+PQ(ω)R)Z=Φ_(C)^(T)L_(M) ⁻¹F_(R) for Z iteratively.
 7. The method of claim 1, furthercomprising: repeatedly solving for Z for each one of a selected set ofexcitation frequencies including the excitation frequency ω.
 8. A methodof solving a first equation associated with a structure, the firstequation being of the form{−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F, wherein ω is atime-harmonic excitation frequency, wherein I₁ is an identity matrixwhich satisfies an equation of the form Φ^(T)MΦ=I₁, wherein M is asymmetric mass matrix, wherein Φ is a matrix having a plurality ofcolumns, each column including an eigenvector approximating one of acorresponding plurality of undamped natural modes of vibrationassociated with the structure satisfying the eigenvalue problem KΦ=MΦΛ,wherein B is a viscous damping matrix, wherein γ is a scalarrepresenting a global structural damping coefficient, wherein Λ is adiagonal matrix which satisfies an equation of the form Φ^(T)KΦ=Λ,wherein K is a symmetric stiffness matrix, wherein K₄ is a symmetricstructural damping matrix representing local departures from the globalstructural damping coefficient γ, wherein Y is a matrix from which anapproximation in the form of a second equation X≈ΦY can be made, andwherein X is a matrix having a plurality of displacement vectors to bedetermined and corresponding to a plurality of load vectors acting onthe structure included in the matrix F, the method comprising: definingan intermediate matrix C=(1+iγ)Λ+i(Φ^(T)K₄Φ); computing a plurality ofeigenvalues and eigenvectors for a third equation CΦ_(C)=Φ_(C)Λ_(C),wherein Φ_(C) is a matrix including eigenvectors of the matrix C, andwherein Λ_(C) is a matrix including eigenvalues of the matrix C;computing a matrix P=Φ_(C) ^(T)Φ^(T)U, and a matrix R=V^(T)ΦΦ_(C),wherein U and V are matrices which satisfy a singular valuedecomposition B=UΣV^(T), wherein Σ is a diagonal matrix includingsingular values; solving a fourth equation of the form{D(ω)+PQ(ω)R}Z=Φ_(C) ^(T)Φ^(T)F for Z, wherein D(ω)=−ω²Φ_(C)^(T)Φ_(C)+Φ_(C) ^(T)CΦ_(C) and Q(ω)=iωΣ; computing Y=Φ_(C) ^(T)Z,wherein the plurality of displacement vectors included in ΦY indicateapproximate measures of displacements in the structure; andcommunicating at least a portion of the plurality of displacementvectors included in ΦY indicating approximate measures of displacementsin the structure to a user interface.
 9. The method of claim 8, furthercomprising: approximately solving a fifth equation of the form{−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the fifth equation byΦ^(T) and solving for a matrix Y, wherein the matrix X is approximatelyequal to ΦY, wherein the matrix Φ is a matrix satisfying the equationsKΦ=MΦΛ, Φ^(T)MΦ=I₃, a third identity matrix, and Φ^(T)KΦ=Λ, wherein thematrix Φ includes a plurality of columns, each column including aneigenvector that approximates one of a corresponding plurality ofundamped natural modes of vibration associated with the structure,wherein Λ is a diagonal matrix including a plurality of eigenvalues,each one of the plurality of eigenvalues corresponding to one of theeigenvectors or columns included in the matrix Φ, and wherein the matrixY includes a plurality of vectors corresponding to the plurality of loadvectors included in the matrix F; and forming an approximation X≈ΦY. 10.The method of claim 9, further comprising: computing the matrix Φ. 11.The method of claim 10, further comprising: providing the matrices M, B,K, K₄, and F, and the scalar γ.
 12. The method of claim 8, furthercomprising: solving the fourth equation of the form {D(ω)+PQ(ω)R}Z=Φ_(C)^(T)Φ^(T)F for Z directly.
 13. The method of claim 8, furthercomprising: solving the fourth equation of the form {D(ω)+PQ(ω)R}Z=Φ_(C)^(T)Φ^(T)F for Z iteratively.
 14. The method of claim 8, furthercomprising: repeatedly solving for Z using each one of a selected set ofexcitation frequencies including the excitation frequency ω.
 15. Amethod of solving a first equation associated with a structure, thefirst equation being of the form{−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F, wherein ω is atime-harmonic excitation frequency, wherein I₁ is an identity matrixwhich satisfies an equation of the form Φ^(T)MΦ=I₁, wherein M is asymmetric mass matrix, wherein Φ is a matrix having a plurality ofcolumns, each column including an eigenvector that approximates one of acorresponding plurality of undamped natural modes of vibrationassociated with the structure, wherein B is a viscous damping matrix,wherein γ is a scalar representing a global structural dampingcoefficient, wherein Λ is a diagonal matrix which satisfies a secondequation of the form Φ^(T)KΦ=Λ, wherein K is a symmetric stiffnessmatrix, wherein K₄ is a symmetric structural damping matrix representinglocal departures from the global structural damping coefficient γ,wherein Y is a matrix which satisfies a third equation of the form X≈ΦY,and wherein X is a matrix having a plurality of displacement vectorscorresponding to a plurality of load vectors acting on the structureincluded in the matrix F, the method comprising: defining anintermediate matrix C=(1+iγ)Λ+i(Φ^(T)K₄Φ); computing a plurality ofeigenvalues and eigenvectors for a fourth equation CΦ_(C)=Φ_(C)Λ_(C),wherein Φ_(C) is a matrix including eigenvectors of the matrix C, andwherein Λ_(C) is a matrix including eigenvalues of the matrix C;computing a matrix P=Φ_(C) ^(T)U_(R), and a matrix R=V_(R) ^(T)Φ_(C),wherein U_(R) and V_(R) are matrices which satisfy a singular valuedecomposition Φ^(T)BΦ=U_(R)Σ_(R)V_(R) ^(T), and wherein ΣR is a diagonalmatrix including singular values; solving a fifth equation of the form{D(ω)+PQ(ω)R}Z=Φ_(C) ^(T)Φ^(T)F for Z, wherein D(ω)=−ω²Φ_(C)^(T)Φ_(C)+Φ_(C) ^(T)CΦ_(C) and Q(ω)=iωΣ_(R); computing Y=Φ_(C) ^(T)Z,wherein the plurality of displacement vectors included in ΦY indicateapproximate measures of displacements in the structure; andcommunicating at least a portion of the plurality of displacementvectors included in ΦY indicating approximate measures of displacementsin the structure to a user interface.
 16. The method of claim 15,further comprising: approximately solving a sixth equation of the form{−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the sixth equation byΦ^(T) and solving for a matrix Y, wherein the matrix X is approximatelyequal to ΦY, wherein the matrix Φ is a matrix satisfying the equationsKΦ=MΦΛ, Φ^(T)MΦ=I₃, a third identity matrix, and Φ^(T)KΦ=Λ, wherein thematrix Φ includes a plurality of columns, each column including aneigenvector that approximates one of a corresponding plurality ofnatural modes of vibration associated with the structure, wherein Λ is adiagonal matrix including a plurality of eigenvalues, each one of theplurality of eigenvalues corresponding to one of the eigenvectors orcolumns included in the matrix Φ, and wherein the matrix Y is a matrixhaving a plurality of vectors corresponding to the plurality of loadvectors included in the matrix F; and forming an approximation X≈ΦY. 17.The method of claim 16, further comprising: computing the matrix Φ. 18.The method of claim 17, further comprising: providing the matrices M, B,K, K₄, and F, and the scalar γ.
 19. The method of claim 15, furthercomprising: solving the fifth equation of the form {D(ω)+PQ(ω)R}Z=Φ_(C)^(T)Φ^(T)F for Z directly.
 20. The method of claim 15, furthercomprising: solving the fifth equation of the form {D(ω)+PQ(ω)R}Z=Φ_(C)^(T)Φ^(T)F for Z iteratively.
 21. The method of claim 15, furthercomprising: repeatedly solving for Z using each one of a selected set ofexcitation frequencies including the excitation frequency ω.
 22. Amethod of solving a first equation associated with a structure, whereinthe first equation is of the form{−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K⁴Φ)]}Y=Φ^(T)F, wherein ω is atime-harmonic excitation frequency, wherein I₁ is an identity matrixwhich satisfies a second equation of the form Φ^(T)MΦ=I₁, wherein M is asymmetric mass matrix, wherein Φ is a matrix having columns spanning asubspace used to approximate a plurality of undamped natural modes ofvibration associated with the structure, wherein B is a viscous dampingmatrix, wherein γ is a scalar representing a global structural dampingcoefficient, wherein Λ is a matrix that satisfies a third equation ofthe form Φ^(T)KΦ=Λ, wherein K is a symmetric stiffness matrix, whereinK₄ is a symmetric structural damping matrix representing localdepartures from the global structural damping coefficient, wherein Y isa matrix from which an approximation in the form of a fourth equationX≈ΦY can be made, and wherein X is a matrix having a plurality ofdisplacement vectors corresponding to a plurality of load vectors actingon the structure included in the matrix F, the method comprising:computing a singular value decomposition for at least one of thematrices B and K₄, wherein B=U_(B)Σ_(B)V_(B) ^(T), and whereinK₄=U_(K4)Σ_(K4)V_(K4) ^(T); generating a matrix P=Φ^(T)[U_(B)U_(K4)] anda matrix R=(Φ^(T)[V_(B)V_(K4)])^(T); solving a fifth equation of theform {D(ω)+PQ(ω)R}Y=Φ^(T)F for Y, wherein D(ω)=−ω²I+(1+iγ)Λ and${Q(\omega)} = {\begin{bmatrix}{{\mathbb{i}}\;\omega\;\Sigma_{B}} & 0 \\0 & {{\mathbb{i}}\;\Sigma_{K4}}\end{bmatrix}.}$ wherein the plurality of displacement vectors includedin ΦY indicate approximate measures of displacements in the structure;and communicating at least a portion of the plurality of displacementvectors included in ΦY indicating approximate measures of displacementsin the structure to a user interface.
 23. The method of claim 22,further comprising: approximately solving a sixth equation of the form{−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the sixth equation byΦ^(T) and solving for a matrix Y, wherein the matrix X is approximatelyequal to ΦY, wherein the matrix Φ is a matrix satisfying the equationsKΦ=MΦΛ, Φ^(T)MΦ=I₂, a second identity matrix, and Φ^(T)KΦ=Λ, wherein thematrix Φ includes a plurality of columns, each column including aneigenvector approximating one of a corresponding plurality of naturalmodes of vibration associated with the structure, wherein Λ is adiagonal matrix including a plurality of eigenvalues, each one of theplurality of eigenvalues corresponding to one of the eigenvectors orcolumns included in the matrix Φ, and wherein the matrix Y is a matrixhaving a plurality of vectors corresponding to the plurality of loadvectors included in the matrix F; and forming an approximation X≈ΦY. 24.The method of claim 23, further comprising: computing the matrix Φ. 25.The method of claim 24, further comprising: providing the matrices M, B,K, K₄, and F, and the scalar γ.
 26. The method of claim 22, furthercomprising: solving the fifth equation of the form {D(ω)+PQ(ω)R}Y=Φ^(T)Ffor Y directly.
 27. The method of claim 22, further comprising: solvingthe fifth equation of the form {D(ω)+PQ(ω)R}Y=Φ^(T)F for Y iteratively.28. The method of claim 22, further comprising: repeatedly solving for Yusing each one of a selected set of excitation frequencies including theexcitation frequency ω.
 29. A method of solving a first equationassociated with a structure, wherein the first equation is of the form{−ω²I₁+iω(Φ^(T)BΦ)+[(1+iγ)Λ+i(Φ^(T)K₄Φ)]}Y=Φ^(T)F, wherein ω is atime-harmonic excitation frequency, wherein I₁ is an identity matrixwhich satisfies an equation of the form Φ^(T)MΦ=I₁, wherein M is asymmetric mass matrix, wherein Φ is a matrix having columns spanning asubspace used to approximate a plurality of undamped natural modes ofvibration associated with the structure, wherein B is a viscous dampingmatrix, wherein γ is a scalar representing a global structural dampingcoefficient, wherein Λ is a matrix which satisfies a second equation ofthe form Φ^(T)KΦ=Λ, wherein K is a symmetric stiffness matrix, whereinK₄ is a symmetric structural damping matrix representing localdepartures from the global structural damping coefficient, wherein Y isa matrix from which an approximation in the form of a third equationX≈ΦY can be made, and wherein X is a matrix having a plurality ofdisplacement vectors corresponding to a plurality of load vectors actingon the structure included in the matrix F, the method comprising:computing a singular value decomposition for at least one of thematrices B_(R)=Φ^(T)BΦ and K_(4R)=Φ^(T)K₄Φ, whereinB_(R)=U_(BR)Σ_(BR)V_(BR) ^(T), and wherein K_(4R)=U_(K4R)Σ_(K4R)V_(K4R)^(T); forming a matrix P=[U_(BR)U_(K4R)] and a matrixR=[V_(BR)V_(K4R)]^(T); solving a fourth equation of the form{D(ω)+PQ(ω)R}Y=Φ^(T)F for Y, wherein D(ω)=−ω²I+(1+iγ)Λ and${Q(\omega)} = {\begin{bmatrix}{{\mathbb{i}}\;\omega\;\Sigma_{B}} & 0 \\0 & {{\mathbb{i}}\;\Sigma_{K4}}\end{bmatrix}.}$ , wherein the plurality of displacement vectorsincluded in ΦY indicate approximate measures of displacements in thestructure; and communicating at least a portion of the plurality ofdisplacement vectors included in ΦY indicating approximate measures ofdisplacements in the structure to a user interface.
 30. The method ofclaim 29, further comprising: approximately solving a fifth equation ofthe form {−ω²M+iωB+[(1+iγ)K+iK₄]}X=F by pre-multiplying the fifthequation by Φ^(T) and solving for a matrix Y, wherein the matrix X isapproximately equal to ΦY, wherein the matrix Φ is a matrix satisfyingthe equations KΦ=MΦΛ, Φ^(T)MΦ=I₂, a second identity matrix, andΦ^(T)KΦ=Λ, wherein the matrix Φ includes a plurality of columns, eachcolumn including an eigenvector that approximates one of a correspondingplurality of natural modes of vibration associated with the structure,wherein Λ is a diagonal matrix including a plurality of eigenvalues,each one of the plurality of eigenvalues corresponding to one of theeigenvectors or columns included in the matrix Φ, and wherein the matrixY is a matrix having a plurality of vectors corresponding to theplurality of load vectors included in the matrix F; and forming anapproximation X≈ΦY.
 31. The method of claim 30, further comprising:computing the matrix Φ.
 32. The method of claim 31, further comprising:providing the matrices M, B, K, K₄, and F, and the scalar γ.
 33. Themethod of claim 29, further comprising: solving the fourth equation ofthe form {D(ω)+PQ(ω)R}Y=Φ^(T)F for Y directly.
 34. The method of claim29, further comprising: solving the fourth equation of the form{D(ω)+PQ(ω)R}Y=Φ^(T)F for Y iteratively.
 35. The method of claim 29,further comprising: repeatedly solving for Y using each one of aselected set of excitation frequencies including the excitationfrequency ω.
 36. An article comprising a machine-accessible mediumhaving associated data used to solve a first equation associated with astructure, the first equation being of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), wherein ω is atime-harmonic excitation frequency, wherein M_(R) is a reduced form of asymmetric mass matrix M, wherein B_(R) is a reduced form of a viscousdamping matrix B, wherein γ is a scalar global structural dampingcoefficient, wherein K_(R) is a reduced form of a symmetric stiffnessmatrix K, wherein K_(4R) is a reduced form of a symmetric structuraldamping matrix K₄ representing local departures from γ, wherein F_(R) isa reduced form of a matrix F including a plurality of load vectorsacting on the structure, and wherein the data, when accessed, results ina machine performing: factoring the reduced matrix M_(R)=L_(M)L_(M)^(T), wherein L_(M) is a lower triangular matrix; defining anintermediate matrix C=(1+iγ)L_(M) ⁻¹K_(R)L_(M) ^(−T)+iL_(M)⁻¹K_(4R)L_(M) ^(−T); computing a plurality of eigenvalues andeigenvectors for a second equation CΦ_(C)=Φ_(C)Λ_(C), wherein Φ_(C) is amatrix including eigenvectors of the matrix C, and wherein Λ_(C) is amatrix including eigenvalues of the matrix C; computing a matrix P=Φ_(C)^(T)L_(M) ⁻¹U_(R) and a matrix R=V_(R) ^(T)L_(M) ^(−T)Φ_(C), whereinU_(R) and V_(R) are matrices which satisfy a singular valuedecomposition B_(R)=U_(R)Σ_(R)V_(R) ^(T), and wherein Σ_(R) is diagonal,conformal with U_(R) and V_(R), and includes singular values; solving athird equation of the form (D(ω)+PQ(ω)R)Z=Φ_(C) ^(T)L_(M) ⁻¹F_(R) for Z,wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ_(R); forming a product Y=L_(M)^(−T)Φ_(C)Z, wherein a matrix Φ comprises eigenvectors satisfying theeigenvalue problem KΦ=MΦΛ, wherein Λ=Φ^(T)KΦ, and wherein the matrix Φmultiplied by the product Y is approximately equal to a matrix Xincluding a plurality of displacements of the structure; andcommunicating at least a portion of the plurality of displacementvectors included in the matrix X indicating approximate measures ofdisplacements in the structure to a user interface.
 37. An apparatus tosolve a first equation associated with a structure, the first equationbeing of the form {−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R),wherein ω is a time-harmonic excitation frequency, wherein M_(R) is areduced form of a symmetric mass matrix M, wherein B_(R) is a reducedform of a viscous damping matrix B, wherein γ is a scalar globalstructural damping coefficient, wherein K_(R) is a reduced form of asymmetric stiffness matrix K, wherein K_(4R) is a reduced form of asymmetric structural damping matrix K₄ representing local departuresfrom γ, and wherein F_(R) is a reduced form of a matrix F including aplurality of load vectors acting on the structure, comprising: afactoring module to factor the reduced matrix M_(R)=L_(M)L_(M) ^(T),wherein L_(M) is a lower triangular; a definition module capable ofbeing communicatively coupled to the factoring module, the definitionmodule to define an intermediate matrix C=(1+iγ)L_(M) ⁻¹K_(R)L_(M)^(−T)+iL_(M) ^(−T)K_(4R)L_(M) ^(−T); a first computation module capableof being communicatively coupled to the definition module, the firstcomputation module to compute a plurality of eigenvalues andeigenvectors for a second equation CΦ_(C)=Φ_(C)Λ_(C), wherein Φ_(C) is amatrix including eigenvectors of the matrix C, and wherein Λ_(C) is amatrix including eigenvalues of the matrix C; a second computationmodule capable of being communicatively coupled to the first computationmodule, the second computation module to compute a matrix P=Φ_(C)^(T)L_(M) ⁻¹U_(R) and a matrix R=V_(R) ^(T)L_(M) ^(−T)Φ_(C), whereinU_(R) and V_(R) are matrices which satisfy a singular valuedecomposition B_(R)=U_(R)Σ_(R)V_(R) ^(T), and wherein Σ_(R) is diagonal,conformal with U_(R) and V_(R), and includes singular values; anequation solving module capable of being communicatively coupled to thesecond computation module, the equation solving module to solve a thirdequation of the form (D(ω)+PQ(ω)R)Z=Φ_(C) ^(T)L_(M) ⁻¹F_(R) for Z,wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ_(R); a product formation module toform a product Y=L_(M) ^(−T)Φ_(C)Z, wherein a matrix Φ compriseseigenvectors satisfying the eigenvalue problem KΦ=MΦΛ, whereinΛ=Φ^(T)KΦ, and wherein the matrix Φ multiplied by the product Y isapproximately equal to a matrix X including a plurality of displacementsof the structure; and a user interface to receive an approximation of avibration-induced displacement of the structure including at least aportion of a product comprising the matrix Φ multiplied by the productY.
 38. A system to solve a first equation associated with a structure,the first equation being of the form{−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R), wherein ω is atime-harmonic excitation frequency, wherein M_(R) is a reduced form of asymmetric mass matrix M, wherein B_(R) is a reduced form of a viscousdamping matrix B, wherein γ is a scalar global structural dampingcoefficient, wherein K_(R) is a reduced form of a symmetric stiffnessmatrix K, wherein K_(4R) is a reduced form of a symmetric structuraldamping matrix K₄ representing local departures from γ, and whereinF_(R) is a reduced form of a matrix F including a plurality of loadvectors acting on the structure, comprising: a processor; a memorycapable of being communicatively coupled to the processor, the memoryincluding a factoring module to factor the reduced matrixM_(R)=L_(M)L_(M) ^(T), wherein L_(M) is a lower triangular matrix; adefinition module capable of being communicatively coupled to thefactoring module, the definition module to define an intermediate matrixC=(1+iγ)L_(M) ⁻¹K_(R)L_(M) ^(−T)+iL_(M) ⁻¹K_(4R)L_(M) ^(−T); a firstcomputation module capable of being communicatively coupled to thedefinition module, the first computation module to compute a pluralityof eigenvalues and eigenvectors for a second equation CΦ_(C)=Φ_(C)Λ_(C),wherein Φ_(C) is a matrix including eigenvectors of the matrix C, andwherein Λ_(C) is a matrix including eigenvalues of the matrix C; asecond computation module capable of being communicatively coupled tothe first computation module, the second computation module to compute amatrix P=Φ_(C) ^(T)L_(M) ⁻¹U_(R) and a matrix R=V_(R) ^(T)L_(M)^(−T)Φ_(C), wherein U_(R) and V_(R) are matrices which satisfy asingular value decomposition B_(R)=U_(R)Σ_(R)V_(R) ^(T), and whereinΣ_(R) is diagonal, conformal with U_(R) and V_(R), and includes singularvalues; an equation solving module capable of being communicativelycoupled to the second computation module, the equation solving module tosolve a third equation of the form (D(ω)+PQ(ω)R)Z=Φ_(C) ^(T)L_(M)⁻¹F_(R) for Z, wherein D(ω)=−ω²I+Λ_(C) and Q(ω)=iωΣ_(R); and a productformation module to form a product Y=L_(M) ^(−T)Φ_(C)Z, wherein a matrixΦ comprises eigenvectors satisfying the eigenvalue problem KΦ=MΦΛ,wherein Λ=Φ^(T)KΦ, and wherein the matrix Φ multiplied by the product Yis approximately equal to a matrix X including a plurality ofdisplacements of the structure; and a user interface to couple to thememory and to receive an approximation of a vibration-induceddisplacement of the structure including at least a portion of a productcomprising the matrix Φ multiplied by the product Y.
 39. A method ofsolving a first equation associated with a structure, the first equationbeing of the form {−ω²M_(R)+iωB_(R)+[(1+iγ)K_(R)+i(K_(4R))]}Y=F_(R),wherein ω is a time-harmonic excitation frequency, wherein M_(R) is areduced form of a symmetric mass matrix M, wherein B_(R) is a reducedform of a viscous damping matrix B, wherein γ is a scalar globalstructural damping coefficient, wherein K_(R) is a reduced form of asymmetric stiffness matrix K, wherein K_(4R) is a reduced form of asymmetric structural damping matrix K₄ representing local departuresfrom γ, and wherein F_(R) is a reduced form of a matrix F including aplurality of load vectors acting on the structure, the methodcomprising: solving an eigenvalue problem of the form[(1+iγ)K_(R)+iK_(4R)]Φ_(G)=M_(R)Φ_(G)Λ_(G); computing a matrix P=Φ_(G)^(T)U_(R) and a matrix R=V_(R) ^(T)Φ_(G), wherein U_(R) and V_(R) arematrices which satisfy a singular value decompositionB_(R)=U_(R)Σ_(R)V_(R) ^(T), and wherein Σ_(R) is diagonal, conformalwith U_(R) and V_(R), and includes singular values; solving a thirdequation of the form (D(ω)+PQ(ω)R)Z=Φ_(G) ^(T)F_(R) for Z, whereinD(ω)=−ω²I+Λ_(G) and Q(ω)=iωΣ_(R), wherein Φ_(G) is normalized to satisfyΦ_(G) ^(T)M_(R)Φ_(G)=I, where I is an identity matrix and Φ_(G)^(T)K_(R)Φ_(G)=Λ_(G); forming a product Y=L_(M) ^(−T)Φ_(G)Z, wherein amatrix Φ comprises eigenvectors satisfying the eigenvalue problemKΦ=MΦΛ, wherein Λ=Φ^(T)KΦ, and wherein the matrix Φ multiplied by theproduct Y is approximately equal to a matrix X including a plurality ofdisplacements of the structure; and communicating an approximation of avibration-induced displacement of the structure, including at least aportion of a product comprising the matrix Φ multiplied by the productY, to a user interface.
 40. A method of solving a first equationassociated with a structure, the first equation being of the form{−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, wherein X is a matrix of displacementvectors, wherein ω is a time-harmonic excitation frequency, wherein M isa symmetric mass matrix, wherein B is a viscous damping matrix, whereinγ is a scalar global structural damping coefficient, wherein K is asymmetric stiffness matrix, wherein K₄ is a symmetric structural dampingmatrix representing local departures from γ, and wherein F is a matrixincluding a plurality of load vectors acting on the structure, themethod comprising: transforming at least two matrices included in a setof matrices comprising M, B, K, K₄ to provide a set of at least twodiagonalized matrices and a set of non-diagonalized matrices, wherein K₄is non-null; forming a matrix D comprising a linear combination of theset of at least two diagonalized matrices; forming a matrix product PQRas a representation of a linear combination of the set ofnon-diagonalized matrices; solving a second equation of the form(D+PQR)Z=A for Z, wherein Z is a frequency response solution matrix, andwherein A is a transformed load matrix; back-transforming the frequencyresponse solution matrix Z to provide the matrix X including thedisplacement vectors indicating measures of displacements in thestructure; and communicating at least a portion of the displacementvectors included in the matrix X indicating measures of displacements inthe structure to a user interface.
 41. The method of claim 40, whereinthe matrices P and R are rectangular, and wherein the matrix Q issquare.
 42. A method of solving a first equation associated with astructure, the first equation being of the form{−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, wherein X is a matrix of displacementvectors, wherein ω is a time-harmonic excitation frequency, wherein M isa symmetric mass matrix, wherein B is a viscous damping matrix, whereinγ is a scalar global structural damping coefficient, wherein K is asymmetric stiffness matrix, wherein K₄ is a symmetric structural dampingmatrix representing local departures from γ, and wherein F is a matrixincluding a plurality of load vectors acting on the structure, themethod comprising: forming a matrix D comprising a linear combination ofa set of at least two diagonal matrices selected from the set ofmatrices M, B, K, K₄, wherein at least two of the matrices M, B, K, K₄are diagonal, and wherein at least one of the matrices B and K₄ isnon-null; forming a matrix product PQR as a representation of a linearcombination of a set of non-diagonal matrices selected from the set ofmatrices M, B, K, K₄; solving a second equation of the form (D+PQR)X=Ffor the matrix X including the displacement vectors indicating measuresof displacements in the structure; and communicating at least a portionof the matrix X including the displacement vectors indicating measuresof displacements in the structure to a user interface.
 43. The method ofclaim 42, wherein the matrices P and R are rectangular, and wherein thematrix Q is square.
 44. A method of approximating a solution of a firstequation associated with a structure, the first equation being of theform {−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, wherein X is a matrix ofdisplacement vectors, wherein ω is a time-harmonic excitation frequency,wherein M is a symmetric mass matrix, wherein B is a viscous dampingmatrix, wherein γ is a scalar global structural damping coefficient,wherein K is a symmetric stiffness matrix, wherein K₄ is a symmetricstructural damping matrix representing local departures from γ, andwherein F is a matrix including a plurality of load vectors acting onthe structure, the method comprising: obtaining reduced forms of thematrices M, B, K, K₄, and F as M_(R)=Φ^(T)MΦ, B_(R)=Φ^(T)BΦ,K_(R)=Φ^(T)KΦ, K_(4R)=Φ^(T)K₄Φ, and F_(R)=Φ^(T)FΦ, respectively, whereinΦ is a matrix having columns spanning a subspace of approximation forapproximating the solution matrix X, and wherein at least one of thematrices B and K₄ is non-null; forming a matrix D comprising a linearcombination of a set of at least two diagonalized matrices selected froma set of matrices obtained from transforming the reduced matrices M_(R),B_(R), K_(R), and K_(4R); forming a matrix product PQR as arepresentation of a linear combination of the set of non-diagonalizedmatrices selected from a set of matrices obtained from transforming thereduced matrices M_(R), B_(R), K_(R), and K_(4R); solving a secondequation of the form (D+PQR)Z=A for Z, wherein Z is a frequency responsesolution matrix, and wherein A is a transformed load matrix;back-transforming the frequency response solution matrix Z to provide aback-transformed matrix Z and multiplying the back-transformed matrix Zby Φ to provide an approximation of the solution matrix X, wherein theapproximation includes a plurality of displacement vectors indicatingapproximate measures of displacements in the structure; andcommunicating at least a portion of the approximation indicatingapproximate measures of displacements in the structure to a userinterface.
 45. The method of claim 44, wherein the matrices P and R arerectangular, and wherein the matrix Q is square.
 46. A method ofapproximating a solution of a first equation associated with astructure, the first equation being of the form{−ω²M+iωB+[(1+iγ)K+i(K₄)]}X=F, wherein X is a matrix of displacementvectors, wherein ω is a time-harmonic excitation frequency, wherein M isa symmetric mass matrix, wherein B is a viscous damping matrix, whereinγ is a scalar global structural damping coefficient, wherein K is asymmetric stiffness matrix, wherein K₄ is a symmetric structural dampingmatrix representing local departures from γ, and wherein F is a matrixincluding a plurality of load vectors acting on the structure, themethod comprising: obtaining reduced forms of the matrices M, B, K, K₄,and F as M_(R)=Φ^(T)MΦ, B_(R)=Φ^(T)BΦ, K_(R)=Φ^(T)KΦ, K_(4R)=Φ^(T)K₄Φ,and F_(R)=Φ^(T)FΦ, respectively, wherein Φ is a matrix having columnsspanning a subspace of approximation for approximating the solutionmatrix X, wherein at least one of the matrices B and K₄ is non-null, andwherein at least two of the matrices selected from a set of reducedmatrices including M_(R), B_(R), K_(R), and K_(4R) are diagonal; forminga matrix D comprising a linear combination of the at least two diagonalmatrices selected from the set of reduced matrices including thematrices M_(R), B_(R), K_(R), and K_(4R); forming a matrix product PQRas a representation of a linear combination of the set of non-diagonalmatrices selected from a set of reduced matrices including the matricesM_(R), B_(R), K_(R), and K_(4R); solving a second equation of the form(D+PQR)Z=F_(R) for Z, wherein Z is a frequency response solution matrix;multiplying the matrix Z by the matrix Φ to provide an approximation ofthe solution matrix X, wherein the approximation includes a plurality ofdisplacement vectors indicating approximate measures of displacements inthe structure; and communicating at least a portion of the approximationindicating approximate measures of displacements in the structure to auser interface.
 47. The method of claim 46, wherein the matrices P and Rare rectangular, and wherein the matrix Q is square.